(Communicated by William D. Sudderth). Abstract. An "exact" ... particular, Maynard [8] provides a complete solution to the problem at issue. De. Finetti [5] .... Second, (x, y/(x)) £ R(m*, p*) in case x = H(t - 0) or x = H(t + 0) for some t. Indeed, take ...
proceedings of the american mathematical society Volume 114, Number 4, APRIL 1992
FINITELY ADDITIVE RADON-NIKODYM THEOREM AND CONCENTRATION FUNCTION OF A PROBABILITY WITH RESPECT TO A PROBABILITY PATRIZIA BERTI, EUGENIO REGAZZINI, AND PIETRO RIGO (Communicated by William D. Sudderth) Abstract. An "exact" Radon-Nikodym theorem is obtained for a pair (m, p.) of finitely-additive probabilities, using a notion of concentration function of p with respect to m . In addition, some direct consequences of that theorem are examined.
0. Introduction Throughout this paper the term probability will designate any positive, finitelyadditive measure on an algebra # of subsets of il, assuming value 1 at il. We turn our attention to the case in which ß and m are probabilities on (il, J), with ß absolutely continuous with respect to (w.r.t.) m [for each e > 0 there exists S > 0 such that ß(E) < e whenever E £ $ and m(E) < ô]. Even if this condition holds, ß need not admit any "exact" Radon-Nikodym derivative w.r.t. m . Actually, under that hypothesis, Bochner [2] states that, given e > 0, there exists a simple function fi, such that | JE fe dm - ß(E)\ < e for all E in $. Later, the same theorem is proved again by various authors at various times (cf. Bhaskara Rao and Bhaskara Rao [1, Chapter 6 and Notes and Comments to Chapter 6]). A few of them deal with the problem of finding necessary and sufficient conditions for the existence of exact Radon-Nikodym derivatives. In particular, Maynard [8] provides a complete solution to the problem at issue. De Finetti [5] already had pointed out the connections between the existence of an exact Radon-Nikodym derivative and certain properties of the lower boundary of the convex hull of the range of (m, ß), under the hypothesis that 5 coincides with the power set of il, and m is strongly continuous [for each e > 0 there exists a partition {E\, ... , E„} of Q in # such that m(E¿) < e for every i]. Recently, de Finetti's approach has been basically taken up again by Received by the editors December 5, 1989.
1980MathematicsSubject Classification(1985 Revision). Primary 60A10, 28A25, 28A60. Key words and phrases. Concentration function, extension, finitely-additive probability, Radon-
Nikodym theorem. The first author's research was partially supported by MPI (40% 1985, Gruppo Nazionale "Processi Stocastici e Calcólo Stocastico"). The second author's research was partially supported by MPI (40% 1987, Gruppo Nazionale "Modelli Probabilistici") and by CNR (Progetto Strategico "Statistica dei Processi Aleatori"). © 1992 American Mathematical Society
0002-9939/92 $1.00+ $.25 per page
1069 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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P. BERTI, E. REGAZZINI, AND P. RIGO
Candeloro and Martellotti [3], supposing that m is strongly continuous, 5 is a cr-algebra, and the exact derivative is picked within the class of real Borelmeasurable functions. In spite of these restrictions, essentially analogous to those originally considered by de Finetti, the above authors have resorted to some quite sophisticated instruments. This paper aims at showing that de Finetti's [5] reasoning can be extended, with slight adjustments, to any pair (m, p) of probabilities, not necessarily strongly continuous, on (il, $). In particular, Candeloro and Martellotti's results follow from ours, which do rely solely on elementary facts. Section 1 includes a few definitions that are used in subsequent sections; specifically, the concept of concentration function is expounded, because it characterizes de Finetti's approach. Section 2 exhibits some properties of the concentration function of ß w.r.t. m when ß possesses an exact derivative w.r.t. m . Section 3 shows that these properties suffice for ß to admit a density w.r.t. m, or at least, w.r.t. some unique extension of m .
1. Preliminaries The first result we quote is the extensibility of any probability from an algebra, of subsets of il, to a larger one. Precisely, suppose m isa probability on an algebra £ and 5* an algebra of subsets of il with 5* d5; then there exists at least one probability m* on 5*, such that m*(A) = m(A) for all A in $. Many of our future developments are based on the notion of integrability in the sense of Dunford and Schwartz [6]. We recall this notion quoting from Bhaskara Rao and Bhaskara Rao [1, Chapter 4]. Let m be a probability on (il, #) and / a simple function with representation / = £)*„, c,7£j, for some real numbers c\, ... , ck and partition {E\, ... , Ek} of il in #. Then, the real number Yl¡=\ c¡m(Ei) is said to be the integral of the simple function f. In order to define integrals of a larger class of functions, one can introduce the concept of hazy convergence of a sequence {fn}^Li of functions from il to R, according to which /„—>/, as n —>oo, if for each e > 0 : inf{m(B): B £$, B D B(n, e)} -> 0 as/?->oo, where B(n, e) = {co £ il: \fn(co) - f(co)\ > e}. Then, a function /: il —» [-oo, oo] is said to be integrable, in the sense of Dunford and Schwartz, if there exists a sequence {fn}%Li of simple functions such that (i) {fn)T=\ converges hazily to / and
(ii) lim„, ^oo fa\fn-fk\dm = 0. If / is integrable, then the integral of f is denoted by Jafdm and is defined to be the number lim^oo Jaf„ dm ; further, the sequence {fn}^Li is called a determining sequence for f. Another notion of great importance for forthcoming developments is that of concentration function. Given any pair of probabilities (m, ß) on (Q, 5)> let R(m, ß) denote its range in R2, R*(m, ß) the convex hull of R(m, ß) and R,(m, ß) the closure of R*(m, p). The concentration function of p w.r.t. m is the function 4> which associates every x in [0, 1] with (¡>(x)- min{y : (x, y) £ R»(m, p)} . Clearly, is continuous, nondecreasing, and convex, with value 0 at 0. Moreover, (f)(1)= I if and only if p is absolutely continuous w.r.t. m. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
FINITELY ADDITIVE RADON-NIKODYM THEOREM
1071
We conclude this section by mentioning a certain subset of [0, 1] associated with (j). Let g be the left-hand derivative of ,J the set of x £ (0, 1) such that g has a jump at x, and G the collection of x £ (0, 1) such that g is strictly increasing at x [i.e. (g(y) - g(x))/(y - x) > 0 whenever y ^ x]. The set:
(1.1)
V =V()= JuGU{l}
will play some role in the characterization ities.
2. Properties
of indefinite integrals w.r.t. probabil-
of the range of (m, p)
WHEN p ADMITS AN EXACT DERIVATIVE WITH RESPECT TO m
In this section we suppose that a nonnegative w-integrable function / exists such that
(2.1)
[ f(co)m(dco) = p(A) for all-in Ja
5,
and we expound a few simple properties of the range of (m, p). Let 5* be any algebra, including #, w.r.t. which / turns out to be Borelmeasurable. After selecting a probability m*, which extends m from # to 5*, define the functions
77(r) = m*({o)£ il: f(to) < t}) q(x) = inf{t£R:H(t)>x}
reR, jce(0,l).
Since f is m-integrable, one can choose a sequence of simple functions, {fn}T=i ' converging hazily to / w.r.t. m (cf. §1). Obviously, {/n}£U converges hazily to / w.r.t. m* as well. Thus, at any continuity point t of 77, the latter can be expressed as
(2.2)
77(i) = lim m({co £ il: fn(co) < t}). n—>oc
In view of (2.2), we are in a position to state If m\ and m2 are extensions of m from # to #* and (2.1) holds, then the set of continuity points of 77i(0 := m\({a> £
(A)
il: f(œ) < t}) and that of H2(t) := m*2({co£ il: f(œ) < t}) coincide, and 77i = 772 on it. Consequently, the quantile function q does not depend on the selected extension of m .
Because of (A), the function
V(x):= / q(y)dy Jo
x£[0,
1]
is, in its turn, independent of the selection of m*. Plainly, y/ is continuous, nondecreasing, and convex, with ^(0) = 0 and y/(\) = fiif(œ)m*(dco) — 1. Our next goal lies in analysing the connections between y/ and the concentration function of p* w.r.t. m*, where p* denotes the extension of p to 5* given
by
(2.3)
p*(E) = [ f(co)m*(da>) Je
E eV ■
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P. BERTI, E. REGAZZINI, AND P. RIGO
1072
First, we point out that (x, y/(x)) £ R(m*, p*) whenever x belongs to the range of 77. In fact, suppose x = 77(0 f°r some t ; then, setting Bx = {cd: f(co) < t} yields m*(Bx) = x and
p*(Bx)= [ fdm*= JBX
i sdH(s)= JO
( [H(t) - H(s)]ds = y/(x). JO
Second, (x, y/(x)) £ R(m*, p*) in case x = H(t - 0) or x = H(t + 0) for some t. Indeed, take a sequence {tn}^Li such that, respectively, tn f t or tn | r, and define B„ = {co: f(co) < t„} and xn = H(t„). Then xn —► x, so that, by continuity of ^, (m*(B„), p*(B„)) = (x„ , y/(x„)) -» (x, yt(x)). Third, for every E £$* and x e (0, 1), m*(E) > x implies /¿*(7?) > ^(x). In fact, let x e [77(i - 0), 77(i + 0)] for some t, and take any sequence {tn}%Lx such that tn f t. Then, again setting xn = H(t„) and B„ = {co: /(tu) < r„} , we obtain
p*(E) = p*(Bn)+ [
JehBc„
fdm*-[
JEcr\B„
fdm*
> p*(Bn) + tnm*(E n Be„)- tnm*(Ec n Bn) = ip(x„) + tn{m*(E) - m*(Bn)} > y/(x„) + t„(x - x„) -* y/(x) as n —>oo.
Let *be the concentration function of p* w.r.t. m*. The third of the above remarks implies that y/(x) < *(x)whenever x is adherent to the range of 77. In particular, ip(x) = *(x)for every x not interior to an interval on which y/ is linear. Consequently, since y/ < 4>*, and since y/ and *are both convex, it must be y/(x) = *(x)for each x in [0, 1]. Now let us turn to the concentration function, ,of p w.r.t. m . If x = 77(r - 0) or x = H(t + 0) for some t, it is always possible to select the above sequence {tn}'%Li in the set of continuity points of 77. In that case, using relation (2.2), for each e > 0 a set B £ 5 can be found such that \m(B) - x| < e and \p(B) - y/(x)\ < e. Consequently (x, y/(x)) £ R(m,p), and since
> (f>*—y/, this proves that = y/ . In short, CR\ *• '
Suppose (2.1) holds and let m* be an extension of m from 5 to 5*; then, assigning p* according to (2.3) implies that *= y/, where (n-\) := Fxatn-\) > i = 1,..., N(n - 1), and are able to apply the previous procedure to each one of the intervals [0, x( 1, n - 1)], [x( 1, n - 1), x(2, w-1)], ... , [x(7V(n—1), n —l), 1] and to the points y¡ —x(i, n), i = 1, ... , N(n), with T^i""1' in the place of Bx for each x in V. Since F^¡~^_x) = License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
FINITELY ADDITIVE RADON-NIKODYM THEOREM
Fx"l,n) >we set: cx(i,n) = F$>H) for i = 1, ... , N(n),
1075
so that {Cx : x £ Zn} D
{Cx: x £ Z„_i}. At this point, let us put, for every neN, E\,n = CX(i;„), Ei>n = CX(i>n)\Cx(i~i,n)
and
for i = 2, ... , N(n)
AT(n)
fn(co) = 5^r(ii=i
1, n)7£/n(tu)
tueíí.
By construction, {/i}£i, is an increasing sequence of simple functions converging to / hazily w.r.t. m, where
(3.2)
f(co) = £(Xö,) ,
xw = inf{x € Z : co £ Cx}
co £ il.
Indeed, if co is in Ex>n, then x,« < x(l, «), so that g(0) < g(xa) < g(x(\, n)) < g(0)+2~n . Otherwise, if tu is in £,;fl , i = 2, ... , N, then xa £ [x(i - 1, n), x(i, n)]. On the other hand, since t(i - 1, n) < g(xw) < t(i, n) whenever x(i - 1, n) < xw < x(i, n), and since t(i, n) - t(i - 1, n) < 2~n , one deduces (N(n)
{co:\f(co)-fn(co)\>2-n}cCcx(Ntn)U
\
|J{tue7ii,„:xCl)
= x(/-l,«)}
.
Let y (A) := inf {m(B): A c B, B £ 5} be the outer measure of A c il associated with m . If co £ Ein and xw = x(i - 1, n), i = 2, ... , N(n), then co must belong to Cx\Cx^-it„) for every x £ Z with x > x(/- 1, n). Moreover, since co £ C¿,-_i>B), the definition of xw implies that there exist points of Z
arbitrarily close to, but strictly greater than, x(i - 1, n). Hence y({co £Eit„:
xw = x(i - 1, «)}) < x - x(i - I, n) for each x > x(i - 1, n), i = 2, ... , N(n)
which gives y({co: \f(co)-fn(co)\>2-"})0,
We will show that JEfndm
-» p(E),
asrc^oo.
for every E in $, as n —»oo. Fix
n £ N, i = 1, ... , N(n), and set x(0, n) = 0, C0 = 0 ; the convexity of implies: (x(i- 1, n)) > (x(i,n)) + g(x(i, n))(x(i— 1, n)-x(i, n)), so that
(3 3)
p(Ei,n) = (p(x(i, n))-(x(i< g(x(i,
Let £65
n))(x(i,
l,n))
n) -x(i-
1, «)) < t(i, n)m(Eiin).
and a := m(Cx{i-Kn) U (En £,-,„)). Then
0(a) < /ivC^-r;») u (B n £,-,„)) = 0(x(i - 1, «)) + /i{£ n 7sf>„) which, denoting the right-hand derivative of 0 by '+and assuming that x(z - 1, n) < x(i, n), yields
r(i - 1, «)m(7i n Eiyn) <
